# American Institute of Mathematical Sciences

November  2017, 37(11): 5883-5912. doi: 10.3934/dcds.2017256

## Visco-Energetic solutions to one-dimensional rate-independent problems

 Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy

Received  October 2016 Revised  June 2017 Published  July 2017

Visco-Energetic solutions of rate-independent systems (recently introduced in [17]) are obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation is reinforced by a viscous correction $δ$, typically a quadratic perturbation of the dissipation distance. Like Energetic and Balanced Viscosity solutions, they provide a variational characterization of rate-independent evolutions, with an accurate description of their jump behaviour.

In the present paper we study Visco-Energetic solutions in the scalar-valued case and we obtain a full characterization for a broad class of energy functionals. In particular, we prove that they exhibit a sort of intermediate behaviour between Energetic and Balanced Viscosity solutions, which can be finely tuned according to the choice of the viscous correction $δ$.

Citation: Luca Minotti. Visco-Energetic solutions to one-dimensional rate-independent problems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5883-5912. doi: 10.3934/dcds.2017256
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##### References:
The double-well potential $W$ with its convex envelope in bold (left picture) and an energetic solution $u$ in the case of a strictly increasing load $\ell$ (right picture).
BV solution for a double-well energy $W$ with an increasing load $\ell$. The blue line denotes the path described by the optimal transition $\vartheta$ solving (4).
Visco-Energetic solutions for a double-well energy $W$ with an increasing load $\ell$. When $\mu>-\min W''$ (left picture) the solution jumps when it reach the maximum of $W'$ and the transition is the ''double chain'' obtained by solving the Incremental Minimization Scheme with frozen time $t$. When $\mu$ is small (right picture) the optimal transition $\vartheta$ makes a first jump connecting $\mathit{u}_{\tiny \mathsf L}(t)$ with $u_+$ according to the modified Maxwell rule (9): $\mathit{u}_{\tiny \mathsf L}(t)$ and $u_+$ corresponds to the intersection of $W'$ with the red line, whose slope is $-\mu$.
The one-sided slopes and the stability region when $W$ is a double-well potential. For some suitable choices of $\delta$, $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ is intermediate between $W'_{\mathsf i\mathsf r}$ and $W'$.
$\mathsf D$-Maxwell's rule for a double well potential: when $\delta=\frac{\mu}{2}|\cdot|^2$, the ''last point'' where $W'$ and $\mathit W_{\mathsf{i}\mathsf{r},{\delta}}'$ coincide is such that the total area between the graph $W'$ and the line whose slope is $-\mu$ is zero.
Visco-Energetic solution of a double-well potential energy with an oscillating external loading and a quadratic viscous-correction $\delta(u,v)$, turned by a parameter $\mu>-\min W''$.
Visco-Energetic solutions of a nonconvex energy and an increasing loading. The optimal transition is a combination of sliding and viscous parts.
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